The use of chaotic systems, particularly in communications, is a rapidly developing field of research. In general, a chaotic system is a dynamical system which has no periodicity and the final state of which depends so sensitively on the system's precise initial state that its time-dependent path is, in effect, long-term unpredictable even though it is deterministic.
One approach to chaotic communication involves a chaotic system controlled by a transmitter/encoder and an identical chaotic system controlled by a receiver/decoder. Communication is divided into two steps: initialization and transmission. The initialization step uses a series of controls to drive the identical chaotic systems in the transmitter/encoder and receiver/decoder into the same periodic state. This is achieved by repeatedly sending a digital initialization stream to both chaotic systems, driving them onto a known, periodically repeating orbit. The necessary digital initialization stream contains less than 16 bits of information. The transmission step then uses a similar series of controls to steer the trajectories of the chaotic system to regions of space that are labeled 0 and 1, corresponding to the plain text of a digital message. In a preferred embodiment, the trajectories move around a two-lobed structure; one lobe is labeled 0, the other 1. The present invention uses the initialization step to produce known periodic orbits on chaotic systems, which are then converted into sounds that approximate traditional music notes.
The ability to drive a chaotic system onto a known periodic orbit, which is a closed loop in 3-dimensional space for a preferred embodiment, provides an entirely new method for music synthesis. By sending a compressed initialization code to the chaotic system, a periodic waveform can be produced that has a rich harmonic structure and sounds musical. The one-dimensional, periodic waveform needed for music applications is achieved by taking the x-, y-, or z-component (or a combination of them) of the periodic orbit over time as the chaotic system evolves. The periodic waveform represents an analog version of a sound, and by sampling the amplitude of the waveform over time, e.g., using audio standard PCM 16, one can produce a digital version of the sound. The harmonic structures of the periodic waveforms are sufficiently varied that they sound like a variety of musical instruments.
Most importantly, the periodic waveforms are produced using a compressed initialization code. Additional bits to determine the frequency and duration of a note to be synthesized are added to the initialization code to produce a compressed control code. In one embodiment of the present invention each note requires a control code of 32 bits of information.
It is an object of the present invention to control a chaotic system to produce musical waveforms. It is a further object to accomplish such control with a compressed initialization code.